question_answer

Tangent and normal at any point P of the parabola meet the axes at T and G respectively, then

A)  \[ST=SG.SP\]     

B)  \[ST\ne SG=SP\]

C)  \[ST=SG\ne SP\]    

D)  \[ST=SG=SP\]

Correct Answer: D

Solution :

 Let a point\[P(a{{t}^{2}},2at)\]on the parabola \[{{y}^{2}}=4ax\] on which the equations of tangent and normal are\[ty=x+a{{t}^{2}}\] and \[y=-tx+2at+a{{t}^{3}}\]respectively. Since, tangent and normal meet the\[x-\]axis at T and G respectively. Therefore, coordinates of T and G are\[(-a{{t}^{2}},0)\]and\[(2a+a{{t}^{2}},0)\]respectively. By the definition of the parabola, \[SP=PM=a+a{{t}^{2}}\] \[SG=VG-VS\] \[=2a+a{{t}^{2}}-a=a+a{{t}^{2}}\] and       \[ST=VS+VT=a+a{{t}^{2}}\] Thus,        \[SP=SG=ST\]